eprop_iaf_psc_delta.h

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/*
 *  eprop_iaf_psc_delta.h
 *
 *  This file is part of NEST.
 *
 *  Copyright (C) 2004 The NEST Initiative
 *
 *  NEST is free software: you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
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#ifndef EPROP_IAF_PSC_DELTA_H
#define EPROP_IAF_PSC_DELTA_H
 
// nestkernel
#include "connection.h"
#include "eprop_archiving_node_impl.h"
#include "eprop_archiving_node_recurrent.h"
#include "eprop_synapse.h"
#include "event.h"
#include "nest_types.h"
#include "ring_buffer.h"
#include "universal_data_logger.h"
 
namespace nest
{
 
/* BeginUserDocs: neuron, e-prop plasticity, current-based, integrate-and-fire, Bellec
 
Short description
+++++++++++++++++
 
Current-based leaky integrate-and-fire neuron model with delta-shaped
postsynaptic currents for e-prop plasticity
 
Description
+++++++++++
 
``eprop_iaf_psc_delta`` is an implementation of a leaky integrate-and-fire
neuron model with delta-shaped postsynaptic currents used for eligibility
propagation (e-prop) plasticity.
 
E-prop plasticity was originally introduced and implemented in TensorFlow in :footcite:p:`Bellec2020`.
 
.. note::
  The neuron dynamics of the ``eprop_iaf_psc_delta`` model (excluding e-prop
  plasticity) are similar to the neuron dynamics of the ``iaf_psc_delta`` model,
  with minor differences, such as the propagator of the post-synaptic current
  and the voltage reset upon a spike.
 
The membrane voltage time course :math:`v_j^t` of the neuron :math:`j` is given by:
 
.. math::
  v_j^t &= \alpha v_j^{t-1} + \sum_{i \neq j} W_{ji}^\text{rec} z_i^{t-1}
    + \sum_i W_{ji}^\text{in} x_i^t \,, \\
  \alpha &= e^{ -\frac{ \Delta t }{ \tau_\text{m} } } \,, \\
 
where :math:`W_{ji}^\text{rec}` and :math:`W_{ji}^\text{in}` are the recurrent and
input synaptic weight matrices, and :math:`z_i^{t-1}` is the recurrent presynaptic
state variable, while :math:`x_i^t` represents the input at time :math:`t`.
 
Descriptions of further parameters and variables can be found in the table below.
 
The spike state variable is expressed by a Heaviside function:
 
.. math::
  z_j^t = H \left( v_j^t - v_\text{th} \right) \,. \\
 
If the membrane voltage crosses the threshold voltage :math:`v_\text{th}`, a spike is
emitted and the membrane voltage is reset to :math:`v_\text{reset}`. After the time step
of the spike emission, the neuron is not able to spike for an absolute refractory period
:math:`t_\text{ref}` during which the membrane potential stays clamped to the reset voltage
:math:`v_\text{reset}`, thus
 
.. math::
  v_m = v_\text{reset} \quad \text{for} \quad t_\text{spk} \leq t \leq t_\text{spk} + t_\text{ref} \,.
 
Spikes arriving while the neuron is refractory are discarded by default. However,
if ``refractory_input`` is set to ``True`` they are damped for each time step
until the end of the refractory period and then added to the membrane voltage.
 
An additional state variable and the corresponding differential equation
represents a piecewise constant external current.
 
See the documentation on the :doc:`iaf_psc_delta<../models/iaf_psc_delta/>` neuron model
for more information on the integration of the subthreshold dynamics.
 
The change of the synaptic weight is calculated from the gradient :math:`g^t` of
the loss :math:`E^t` with respect to the synaptic weight :math:`W_{ji}`:
:math:`\frac{ \text{d} E^t }{ \text{d} W_{ij} }`
which depends on the presynaptic
spikes :math:`z_i^{t-2}`, the surrogate gradient or pseudo-derivative
of the spike state variable with respect to the postsynaptic membrane
voltage :math:`\psi_j^{t-1}` (the product of which forms the eligibility
trace :math:`e_{ji}^{t-1}`), and the learning signal :math:`L_j^t` emitted
by the readout neurons.
 
Surrogate gradients help overcome the challenge of the spiking function's
non-differentiability, facilitating the use of gradient-based learning
techniques such as e-prop. The non-existent derivative of the spiking
variable with respect to the membrane voltage,
:math:`\frac{\partial z^t_j}{ \partial v^t_j}`, can be effectively
replaced with a variety of surrogate gradient functions, as detailed in
various studies (see, e.g., :footcite:p:`Neftci2019`). NEST currently provides four
different surrogate gradient functions:
 
1. A piecewise linear function used among others in :footcite:p:`Bellec2020`:
 
.. math::
  \psi_j^t = \frac{ \gamma }{ v_\text{th} } \text{max}
    \left( 0, 1-\beta \left| \frac{ v_j^t - v_\text{th} }{ v_\text{th} }\right| \right) \,. \\
 
2. An exponential function used in :footcite:p:`Shrestha2018`:
 
.. math::
  \psi_j^t = \gamma \exp \left( -\beta \left| v_j^t - v_\text{th} \right| \right) \,. \\
 
3. The derivative of a fast sigmoid function used in :footcite:p:`Zenke2018`:
 
.. math::
  \psi_j^t = \gamma \left( 1 + \beta \left| v_j^t - v_\text{th} \right| \right)^2 \,. \\
 
4. The derivative of an arctan function used in :footcite:p:`Fang2021`:
 
.. math::
  \psi_j^t = \frac{\gamma}{\pi} \frac{1}{ 1 + \left( \beta \pi \left( v_j^t - v_\text{th} \right) \right)^2 } \,. \\
 
In the interval between two presynaptic spikes, the gradient is calculated
at each time step until the cutoff time point. This computation occurs over
the time range:
 
:math:`t \in \left[ t_\text{spk,prev}, \min \left( t_\text{spk,prev} + \Delta t_\text{c}, t_\text{spk,curr} \right)
\right]`.
 
Here, :math:`t_\text{spk,prev}` represents the time of the previous spike that
passed the synapse, while :math:`t_\text{spk,curr}` is the time of the
current spike, which triggers the application of the learning rule and the
subsequent synaptic weight update. The cutoff :math:`\Delta t_\text{c}`
defines the maximum allowable interval for integration between spikes.
The expression for the gradient is given by:
 
.. math::
  \frac{ \text{d} E^t }{ \text{d} W_{ji} } &= L_j^t \bar{e}_{ji}^{t-1} \,, \\
  e_{ji}^{t-1} &= \psi_j^{t-1} \bar{z}_i^{t-2} \,, \\
 
The eligibility trace and the presynaptic spike trains are low-pass filtered
with the following exponential kernels:
 
.. math::
  \bar{e}_{ji}^t &= \mathcal{F}_\kappa \left( e_{ji}^t \right)
    = \kappa \bar{e}_{ji}^{t-1} + \left( 1 - \kappa \right) e_{ji}^t \,, \\
  \bar{z}_i^t &= \mathcal{F}_\alpha \left( z_{i}^t \right)= \alpha \bar{z}_i^{t-1} + z_i^t \,. \\
 
Furthermore, a firing rate regularization mechanism keeps the exponential moving average of the postsynaptic
neuron's firing rate :math:`f_j^{\text{ema},t}` close to a target firing rate
:math:`f^\text{target}`. The gradient :math:`g_\text{reg}^t` of the regularization loss :math:`E_\text{reg}^t`
with respect to the synaptic weight :math:`W_{ji}` is given by:
 
.. math::
  \frac{ \text{d} E_\text{reg}^t }{ \text{d} W_{ji}}
    &\approx c_\text{reg} \left( f^{\text{ema},t}_j - f^\text{target} \right) \bar{e}_{ji}^t \,, \\
  f^{\text{ema},t}_j &= \mathcal{F}_{\kappa_\text{reg}} \left( \frac{z_j^t}{\Delta t} \right)
    = \kappa_\text{reg} f^{\text{ema},t-1}_j + \left( 1 - \kappa_\text{reg} \right) \frac{z_j^t}{\Delta t} \,, \\
 
where :math:`c_\text{reg}` is a constant scaling factor.
 
The overall gradient is given by the addition of the two gradients.
 
As a last step for every round in the loop over the time steps :math:`t`, the new weight is retrieved by feeding the
current gradient :math:`g^t` to the optimizer (see :doc:`weight_optimizer<../models/weight_optimizer/>`
for more information on the available optimizers):
 
.. math::
  w^t = \text{optimizer} \left( t, g^t, w^{t-1} \right) \,. \\
 
After the loop has terminated, the filtered dynamic variables of e-prop are propagated from the end of the cutoff until
the next spike:
 
.. math::
  p &= \text{max} \left( 0, t_\text{s}^{t} - \left( t_\text{s}^{t-1} + {\Delta t}_\text{c} \right) \right) \,, \\
  \bar{e}_{ji}^{t+p} &= \bar{e}_{ji}^t \kappa^p \,, \\
  \bar{z}_i^{t+p} &= \bar{z}_i^t \alpha^p \,. \\
 
For more information on the implementation details of the neuron model, see :footcite:p:`Rotter1999` and
:footcite:p:`Diesmann2001`.
 
For more information on e-prop plasticity, see the documentation on the other e-prop models:
 
 * :doc:`eprop_iaf_psc_delta_adapt<../models/eprop_iaf_psc_delta_adapt/>`
 * :doc:`eprop_readout<../models/eprop_readout/>`
 * :doc:`eprop_synapse<../models/eprop_synapse/>`
 * :doc:`eprop_learning_signal_connection<../models/eprop_learning_signal_connection/>`
 
Details on the event-based NEST implementation of e-prop can be found in :footcite:p:`KorcsakGorzo2025`.
 
Parameters
++++++++++
 
The following parameters can be set in the status dictionary.
 
=========================== ======= ======================= ================ ===================================
**Neuron parameters**
----------------------------------------------------------------------------------------------------------------
Parameter                   Unit    Math equivalent         Default          Description
=========================== ======= ======================= ================ ===================================
``C_m``                     pF      :math:`C_\text{m}`                 250.0 Capacitance of the membrane
``E_L``                     mV      :math:`E_\text{L}`                 -70.0 Leak / resting membrane potential
``I_e``                     pA      :math:`I_\text{e}`                   0.0 Constant external input current
``t_ref``                   ms      :math:`t_\text{ref}`                 2.0 Duration of the refractory period
``tau_m``                   ms      :math:`\tau_\text{m}`               10.0 Time constant of the membrane
``V_min``                   mV      :math:`v_\text{min}`    negative maximum Absolute lower bound of the
                                                            value            membrane voltage
                                                            representable
                                                            by ``double``
                                                            type in C++
``V_th``                    mV      :math:`v_\text{th}`                -55.0 Spike threshold voltage
``V_reset``                 mV      :math:`v_\text{reset}`             -70.0 Reset voltage
``refractory_input``        Boolean                                ``False`` If ``True``, spikes arriving during
                                                                             the refractory period are damped
                                                                             until it ends and then added to the
                                                                             membrane voltage
=========================== ======= ======================= ================ ===================================
 
=============================== ======= =========================== ================== =========================
**E-prop parameters**
----------------------------------------------------------------------------------------------------------------
Parameter                       Unit    Math equivalent             Default            Description
=============================== ======= =========================== ================== =========================
``flush_event_send_interval``   ms                                  maximum value      Interval since previous
                                                                    representable by   event after which a flush
                                                                    ``double`` type in event is sent
                                                                    C++
``c_reg``                               :math:`c_\text{reg}`                       0.0 Coefficient of firing
                                                                                       rate regularization
``eprop_isi_trace_cutoff``      ms      :math:`{\Delta t}_\text{c}` maximum value      Cutoff for integration of
                                                                    representable      e-prop update between two
                                                                    by ``double``      spikes
                                                                    type in C++
``f_target``                    Hz      :math:`f^\text{target}`                   10.0 Target firing rate of
                                                                                       rate regularization
``kappa``                               :math:`\kappa`                            0.97 Low-pass filter of the
                                                                                       eligibility trace
``kappa_reg``                           :math:`\kappa_\text{reg}`                 0.97 Low-pass filter of the
                                                                                       firing rate for
                                                                                       regularization
``surrogate_gradient_function``         :math:`\psi`                "piecewise_linear" Surrogate gradient /
                                                                                       pseudo-derivative
                                                                                       function
                                                                                       ["piecewise_linear",
                                                                                       "exponential",
                                                                                       "fast_sigmoid_derivative"
                                                                                       , "arctan_derivative"]
``surrogate_gradient_height``           :math:`\gamma`                             0.3 Height scaling of
                                                                                       surrogate gradient /
                                                                                       pseudo-derivative of
                                                                                       membrane voltage
``surrogate_gradient_width``            :math:`1/\beta`                            1.0 Width scaling of
                                                                                       surrogate gradient /
                                                                                       pseudo-derivative of
                                                                                       membrane voltage
=============================== ======= =========================== ================== =========================
 
Recordables
+++++++++++
 
The following state variables evolve during simulation and can be recorded.
 
================== ==== =============== ============= ========================
**Neuron state variables and recordables**
------------------------------------------------------------------------------
State variable     Unit Math equivalent Initial value Description
================== ==== =============== ============= ========================
``V_m``              mV :math:`v_j`             -70.0 Membrane voltage
================== ==== =============== ============= ========================
 
====================== ==== =============== ============= =========================================
**E-prop state variables and recordables**
---------------------------------------------------------------------------------------------------
State variable         Unit Math equivalent Initial value Description
====================== ==== =============== ============= =========================================
``learning_signal``      pA :math:`L_j`               0.0 Learning signal
``surrogate_gradient``      :math:`\psi_j`            0.0 Surrogate gradient / pseudo-derivative of
                                                          membrane voltage
====================== ==== =============== ============= =========================================
 
Usage
+++++
 
This model can only be used in combination with the other e-prop models
and the network architecture requires specific wiring, input, and output.
The usage is demonstrated in several
:doc:`supervised regression and classification tasks <../auto_examples/eprop_plasticity/index>`
reproducing among others the original proof-of-concept tasks in :footcite:p:`Bellec2020`.
 
References
++++++++++
 
.. footbibliography::
 
Sends
+++++
 
SpikeEvent
 
Receives
++++++++
 
SpikeEvent, CurrentEvent, LearningSignalConnectionEvent, DataLoggingRequest
 
See also
++++++++
 
Examples using this model
+++++++++++++++++++++++++
 
.. listexamples:: eprop_iaf_psc_delta
 
EndUserDocs */
 
void register_eprop_iaf_psc_delta( const std::string& name );
 
class eprop_iaf_psc_delta : public EpropArchivingNodeRecurrent< false >
{
 
public:
  eprop_iaf_psc_delta();
 
  eprop_iaf_psc_delta( const eprop_iaf_psc_delta& );
 
  using Node::handle;
  using Node::handles_test_event;
 
  size_t send_test_event( Node&, size_t, synindex, bool ) override;
 
  void handle( SpikeEvent& ) override;
  void handle( CurrentEvent& ) override;
  void handle( LearningSignalConnectionEvent& ) override;
  void handle( DataLoggingRequest& ) override;
 
  size_t handles_test_event( SpikeEvent&, size_t ) override;
  size_t handles_test_event( CurrentEvent&, size_t ) override;
  size_t handles_test_event( LearningSignalConnectionEvent&, size_t ) override;
  size_t handles_test_event( DataLoggingRequest&, size_t ) override;
 
  void get_status( Dictionary& ) const override;
  void set_status( const Dictionary& ) override;
 
private:
  void init_buffers_() override;
  void pre_run_hook() override;
 
  void update( Time const&, const long, const long ) override;
 
  void compute_gradient( const long,
    const long,
    double&,
    double&,
    double&,
    double&,
    double&,
    double&,
    const CommonSynapseProperties&,
    WeightOptimizer*,
    const bool,
    const bool,
    double&,
    long&,
    long& ) override;
 
  long get_shift() const override;
  bool is_eprop_recurrent_node() const override;
 
  friend class RecordablesMap< eprop_iaf_psc_delta >;
 
  friend class UniversalDataLogger< eprop_iaf_psc_delta >;
 
  struct Parameters_
  {
    double tau_m_;
 
    double C_m_;
 
    double t_ref_;
 
    double E_L_;
 
    double I_e_;
 
    double V_th_;
 
    double V_min_;
 
    double V_reset_;
 
    bool with_refr_input_;
 
    double c_reg_;
 
    double f_target_;
 
    std::string surrogate_gradient_function_;
 
    double surrogate_gradient_height_;
 
    double surrogate_gradient_width_;
 
    double kappa_;
 
    double kappa_reg_;
 
    Parameters_();
 
    void get( Dictionary& ) const;
 
    double set( const Dictionary&, Node* );
  };
 
  struct State_
  {
    double i_in_;
 
    double v_m_;
 
    long r_;
 
    double refr_spikes_buffer_;
 
    double learning_signal_;
 
    double surrogate_gradient_;
 
    State_();
 
    void get( Dictionary&, const Parameters_& ) const;
 
    void set( const Dictionary&, const Parameters_&, double, Node* );
  };
 
  struct Buffers_
  {
    Buffers_( eprop_iaf_psc_delta& );
 
    Buffers_( const Buffers_&, eprop_iaf_psc_delta& );
 
    RingBuffer spikes_;
 
    RingBuffer currents_;
 
    UniversalDataLogger< eprop_iaf_psc_delta > logger_;
  };
 
  struct Variables_
  {
    double P_v_m_;
 
    double P_i_in_;
 
    long RefractoryCounts_;
  };
 
  double
  get_v_m_() const
  {
    return S_.v_m_ + P_.E_L_;
  }
 
  double
  get_surrogate_gradient_() const
  {
    return S_.surrogate_gradient_;
  }
 
  double
  get_learning_signal_() const
  {
    return S_.learning_signal_;
  }
 
  // the order in which the structure instances are defined is important for speed
 
  Parameters_ P_;
 
  State_ S_;
 
  Variables_ V_;
 
  Buffers_ B_;
 
  static RecordablesMap< eprop_iaf_psc_delta > recordablesMap_;
};
 
inline long
eprop_iaf_psc_delta::get_shift() const
{
  return offset_gen_ + delay_in_rec_;
}
 
inline bool
eprop_iaf_psc_delta::is_eprop_recurrent_node() const
{
  return true;
}
 
inline size_t
eprop_iaf_psc_delta::send_test_event( Node& target, size_t receptor_type, synindex, bool )
{
  SpikeEvent e;
  e.set_sender( *this );
  return target.handles_test_event( e, receptor_type );
}
 
inline size_t
eprop_iaf_psc_delta::handles_test_event( SpikeEvent&, size_t receptor_type )
{
  if ( receptor_type != 0 )
  {
    throw UnknownReceptorType( receptor_type, get_name() );
  }
 
  return 0;
}
 
inline size_t
eprop_iaf_psc_delta::handles_test_event( CurrentEvent&, size_t receptor_type )
{
  if ( receptor_type != 0 )
  {
    throw UnknownReceptorType( receptor_type, get_name() );
  }
 
  return 0;
}
 
inline size_t
eprop_iaf_psc_delta::handles_test_event( LearningSignalConnectionEvent&, size_t receptor_type )
{
  if ( receptor_type != 0 )
  {
    throw UnknownReceptorType( receptor_type, get_name() );
  }
 
  return 0;
}
 
inline size_t
eprop_iaf_psc_delta::handles_test_event( DataLoggingRequest& dlr, size_t receptor_type )
{
  if ( receptor_type != 0 )
  {
    throw UnknownReceptorType( receptor_type, get_name() );
  }
 
  return B_.logger_.connect_logging_device( dlr, recordablesMap_ );
}
 
inline void
eprop_iaf_psc_delta::get_status( Dictionary& d ) const
{
  EpropArchivingNodeRecurrent::get_status( d );
  P_.get( d );
  S_.get( d, P_ );
  d[ names::recordables ] = recordablesMap_.get_list();
}
 
inline void
eprop_iaf_psc_delta::set_status( const Dictionary& d )
{
  EpropArchivingNodeRecurrent::set_status( d );
  // temporary copies in case of errors
  Parameters_ ptmp = P_;
  State_ stmp = S_;
 
  // make sure that ptmp and stmp consistent - throw BadProperty if not
  const double delta_EL = ptmp.set( d, this );
  stmp.set( d, ptmp, delta_EL, this );
 
  P_ = ptmp;
  S_ = stmp;
}
 
}  // namespace nest
 
#endif  // EPROP_IAF_PSC_DELTA_H